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Source code: Lib/random.py
Almost all module functions depend on the basic function random(), which
generates a random float uniformly in the half-open range 0.0 <= X < 1.0.
Python uses the Mersenne Twister as the core generator. It produces 53-bit precision
floats and has a period of 2**19937-1. The underlying implementation in C is
both fast and threadsafe. The Mersenne Twister is one of the most extensively
tested random number generators in existence. However, being completely
deterministic, it is not suitable for all purposes, and is completely unsuitable
for cryptographic purposes.
The functions supplied by this module are actually bound methods of a hidden instance of the random.Random class. You can instantiate your own instances of Random to get generators that don’t share state.
Class Random can also be subclassed if you want to use a different basic generator of your own devising: see the documentation on that class for more details.
The random module also provides the SystemRandom class which uses the system function os.urandom() to generate random numbers from sources provided by the operating system.
Warning: The pseudo-random generators of this module should not be used for security purposes. For security or cryptographic uses, see the secrets module.
See also: M. Matsumoto and T. Nishimura, “Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator”, ACM Transactions on Modeling and Computer Simulation Vol. 8, No. 1, January pp.3–30 1998.
Complementary-Multiply-with-Carry recipe for a compatible alternative random number generator with a long period and comparatively simple update operations.
If a is omitted or None, the current system time is used. If
randomness sources are provided by the operating system, they are used
instead of the system time (see the os.urandom() function for details
on availability).
Return an object capturing the current internal state of the generator. This object can be passed to setstate() to restore the state.
state should have been obtained from a previous call to getstate(), and setstate() restores the internal state of the generator to what it was at the time getstate() was called.
This method should not be used for generating security tokens. Use secrets.token_bytes() instead.
The positional argument pattern matches the range() function.
Changed in version 3.2: randrange() is more sophisticated about producing equally distributed
values. Formerly it used a style like int(random()*n) which could produce
slightly uneven distributions.
Changed in version 3.12: Automatic conversion of non-integer types is no longer supported.
Calls such as randrange(10.0) and randrange(Fraction(10, 1))
now raise a TypeError.
Returns a non-negative Python integer with k random bits. This method is supplied with the Mersenne Twister generator and some other generators may also provide it as an optional part of the API. When available, getrandbits() enables randrange() to handle arbitrarily large ranges.
Return a random element from the non-empty sequence seq. If seq is empty, raises IndexError.
Return a k sized list of elements chosen from the population with replacement. If the population is empty, raises IndexError.
If a weights sequence is specified, selections are made according to the
relative weights. Alternatively, if a cum_weights sequence is given, the
selections are made according to the cumulative weights (perhaps computed
using itertools.accumulate()). For example, the relative weights
[10, 5, 30, 5] are equivalent to the cumulative weights
[10, 15, 45, 50]. Internally, the relative weights are converted to
cumulative weights before making selections, so supplying the cumulative
weights saves work.
If neither weights nor cum_weights are specified, selections are made with equal probability. If a weights sequence is supplied, it must be the same length as the population sequence. It is a TypeError to specify both weights and cum_weights.
The weights or cum_weights can use any numeric type that interoperates with the float values returned by random() (that includes integers, floats, and fractions but excludes decimals). Weights are assumed to be non-negative and finite. A ValueError is raised if all weights are zero.
For a given seed, the choices() function with equal weighting typically produces a different sequence than repeated calls to choice(). The algorithm used by choices() uses floating-point arithmetic for internal consistency and speed. The algorithm used by choice() defaults to integer arithmetic with repeated selections to avoid small biases from round-off error.
Changed in version 3.9: Raises a ValueError if all weights are zero.
Note that even for small len(x), the total number of permutations of x
can quickly grow larger than the period of most random number generators.
This implies that most permutations of a long sequence can never be
generated. For example, a sequence of length 2080 is the largest that
can fit within the period of the Mersenne Twister random number generator.
Returns a new list containing elements from the population while leaving the original population unchanged. The resulting list is in selection order so that all sub-slices will also be valid random samples. This allows raffle winners (the sample) to be partitioned into grand prize and second place winners (the subslices).
Members of the population need not be hashable or unique. If the population contains repeats, then each occurrence is a possible selection in the sample.
To choose a sample from a range of integers, use a range() object as an
argument. This is especially fast and space efficient for sampling from a large
population: sample(range(10000000), k=60).
If the sample size is larger than the population size, a ValueError is raised.
Binomial distribution. Return the number of successes for n independent trials with the probability of success in each trial being p:
Exponential distribution. lambd is 1.0 divided by the desired mean. It should be nonzero. (The parameter would be called “lambda”, but that is a reserved word in Python.) Returned values range from 0 to positive infinity if lambd is positive, and from negative infinity to 0 if lambd is negative.
Normal distribution, also called the Gaussian distribution. mu is the mean, and sigma is the standard deviation. This is slightly faster than the normalvariate() function defined below.
Multithreading note: When two threads call this function simultaneously, it is possible that they will receive the same return value. This can be avoided in three ways.
- Have each thread use a different instance of the random number generator. 2) Put locks around all calls. 3) Use the slower, but thread-safe normalvariate() function instead.
Class that implements the default pseudo-random number generator used by the random module.
Subclasses of Random should override the following methods if they
wish to make use of a different basic generator:
Override this method in subclasses to customise the getstate()
behaviour of Random instances.
Override this method in subclasses to customise the setstate()
behaviour of Random instances.
Override this method in subclasses to customise the
getrandbits() behaviour of Random instances.
Class that uses the os.urandom() function for generating random numbers from sources provided by the operating system. Not available on all systems. Does not rely on software state, and sequences are not reproducible. Accordingly, the seed() method has no effect and is ignored. The getstate() and setstate() methods raise NotImplementedError if called.
- If a new seeding method is added, then a backward compatible seeder will be offered.
- The generator’s random() method will continue to produce the same sequence when the compatible seeder is given the same seed.
>>> random() # Random float: 0.0 <= x < 1.0
0.37444887175646646
>>> uniform(2.5, 10.0) # Random float: 2.5 <= x <= 10.0
3.1800146073117523
>>> expovariate(1 / 5) # Interval between arrivals averaging 5 seconds
5.148957571865031
>>> randrange(10) # Integer from 0 to 9 inclusive
7
>>> randrange(0, 101, 2) # Even integer from 0 to 100 inclusive
26
>>> choice(['win', 'lose', 'draw']) # Single random element from a sequence
'draw'
>>> deck = 'ace two three four'.split()
>>> shuffle(deck) # Shuffle a list
>>> deck
['four', 'two', 'ace', 'three']
>>> sample([10, 20, 30, 40, 50], k=4) # Four samples without replacement
[40, 10, 50, 30]
>>> # Six roulette wheel spins (weighted sampling with replacement)
>>> choices(['red', 'black', 'green'], [18, 18, 2], k=6)
['red', 'green', 'black', 'black', 'red', 'black']
>>> # Deal 20 cards without replacement from a deck
>>> # of 52 playing cards, and determine the proportion of cards
>>> # with a ten-value: ten, jack, queen, or king.
>>> deal = sample(['tens', 'low cards'], counts=[16, 36], k=20)
>>> deal.count('tens') / 20
0.15
>>> # Estimate the probability of getting 5 or more heads from 7 spins
>>> # of a biased coin that settles on heads 60% of the time.
>>> sum(binomialvariate(n=7, p=0.6) >= 5 for i in range(10_000)) / 10_000
0.4169
>>> # Probability of the median of 5 samples being in middle two quartiles
>>> def trial():
... return 2_500 <= sorted(choices(range(10_000), k=5))[2] < 7_500
...
>>> sum(trial() for i in range(10_000)) / 10_000
0.7958
Example of statistical bootstrapping using resampling with replacement to estimate a confidence interval for the mean of a sample:
# https://www.thoughtco.com/example-of-bootstrapping-3126155
from statistics import fmean as mean
from random import choices
data = [41, 50, 29, 37, 81, 30, 73, 63, 20, 35, 68, 22, 60, 31, 95]
means = sorted(mean(choices(data, k=len(data))) for i in range(100))
print(f'The sample mean of {mean(data):.1f} has a 90% confidence '
f'interval from {means[5]:.1f} to {means[94]:.1f}')
Example of a resampling permutation test to determine the statistical significance or p-value of an observed difference between the effects of a drug versus a placebo:
# Example from "Statistics is Easy" by Dennis Shasha and Manda Wilson
from statistics import fmean as mean
from random import shuffle
drug = [54, 73, 53, 70, 73, 68, 52, 65, 65]
placebo = [54, 51, 58, 44, 55, 52, 42, 47, 58, 46]
observed_diff = mean(drug) - mean(placebo)
n = 10_000
count = 0
combined = drug + placebo
for i in range(n):
shuffle(combined)
new_diff = mean(combined[:len(drug)]) - mean(combined[len(drug):])
count += (new_diff >= observed_diff)
print(f'{n} label reshufflings produced only {count} instances with a difference')
print(f'at least as extreme as the observed difference of {observed_diff:.1f}.')
print(f'The one-sided p-value of {count / n:.4f} leads us to reject the null')
print(f'hypothesis that there is no difference between the drug and the placebo.')
from heapq import heapify, heapreplace
from random import expovariate, gauss
from statistics import mean, quantiles
average_arrival_interval = 5.6
average_service_time = 15.0
stdev_service_time = 3.5
num_servers = 3
waits = []
arrival_time = 0.0
servers = [0.0] * num_servers # time when each server becomes available
heapify(servers)
for i in range(1_000_000):
arrival_time += expovariate(1.0 / average_arrival_interval)
next_server_available = servers[0]
wait = max(0.0, next_server_available - arrival_time)
waits.append(wait)
service_duration = max(0.0, gauss(average_service_time, stdev_service_time))
service_completed = arrival_time + wait + service_duration
heapreplace(servers, service_completed)
print(f'Mean wait: {mean(waits):.1f} Max wait: {max(waits):.1f}')
print('Quartiles:', [round(q, 1) for q in quantiles(waits)])
See also: Statistics for Hackers a video tutorial by Jake Vanderplas on statistical analysis using just a few fundamental concepts including simulation, sampling, shuffling, and cross-validation.
Economics Simulation a simulation of a marketplace by Peter Norvig that shows effective use of many of the tools and distributions provided by this module (gauss, uniform, sample, betavariate, choice, triangular, and randrange).
A Concrete Introduction to Probability (using Python) a tutorial by Peter Norvig covering the basics of probability theory, how to write simulations, and how to perform data analysis using Python.
These recipes show how to efficiently make random selections from the combinatoric iterators in the itertools module:
def random_product(*args, repeat=1):
"Random selection from itertools.product(*args, **kwds)"
pools = [tuple(pool) for pool in args] * repeat
return tuple(map(random.choice, pools))
def random_permutation(iterable, r=None):
"Random selection from itertools.permutations(iterable, r)"
pool = tuple(iterable)
r = len(pool) if r is None else r
return tuple(random.sample(pool, r))
def random_combination(iterable, r):
"Random selection from itertools.combinations(iterable, r)"
pool = tuple(iterable)
n = len(pool)
indices = sorted(random.sample(range(n), r))
return tuple(pool[i] for i in indices)
def random_combination_with_replacement(iterable, r):
"Choose r elements with replacement. Order the result to match the iterable."
# Result will be in set(itertools.combinations_with_replacement(iterable, r)).
pool = tuple(iterable)
n = len(pool)
indices = sorted(random.choices(range(n), k=r))
return tuple(pool[i] for i in indices)
The default random() returns multiples of 2⁻⁵³ in the range
0.0 ≤ x < 1.0. All such numbers are evenly spaced and are exactly
representable as Python floats. However, many other representable
floats in that interval are not possible selections. For example,
0.05954861408025609 isn’t an integer multiple of 2⁻⁵³.
The following recipe takes a different approach. All floats in the interval are possible selections. The mantissa comes from a uniform distribution of integers in the range 2⁵² ≤ mantissa < 2⁵³. The exponent comes from a geometric distribution where exponents smaller than -53 occur half as often as the next larger exponent.
All real valued distributions in the class will use the new method:
The recipe is conceptually equivalent to an algorithm that chooses from
all the multiples of 2⁻¹⁰⁷⁴ in the range 0.0 ≤ x < 1.0. All such
numbers are evenly spaced, but most have to be rounded down to the
nearest representable Python float. (The value 2⁻¹⁰⁷⁴ is the smallest
positive unnormalized float and is equal to math.ulp(0.0).)
See also: Generating Pseudo-random Floating-Point Values a paper by Allen B. Downey describing ways to generate more fine-grained floats than normally generated by random().